Countable - Remarkably It Works., Are All Infinite Sets Countable?

natural paired pairing elements

Every set that can be counted is countable, but this is no surprise. The interesting case for countable sets comes when we abandon finite sets and consider infinite ones.

An infinite set of numbers, points, or other elements is said to be "countable" (also called denumerable) if its elements can be paired one-to-one with the natural numbers, 1, 2, 3, etc. The term countable is somewhat misleading because, of course, it is humanly impossible actually to count infinitely many things.

The set of even numbers is an example of a countable set, as the pairing in Table 1 shows.

Of course, it is not enough to show the way the first eight numbers are to be paired. One must show that no matter how far one goes along the list of even natural numbers there is a natural number paired with it. In this case this is an easy thing to do. One simply pairs any even number 2n with the natural number n.

What about the set of integers? One might guess that it is uncountable because the set of natural numbers is a proper subset of it. Consider the pairing in Table 2.

Additional Topics

The secret in finding this pairing was to avoid a trap. Had the pairing been that which appears in Table 3, one would never reach the negative integers. In working with infinite sets, one considers a pairing complete if there is a scheme that enables one to reach any number or element in the set after a finite number of steps. (Not everyone agrees that that is the same thing as reaching them all.)…

The answer to this question was given around 1870 by the German mathematician George Cantor. He showed that the set of numbers between 0 and 1 represented by infinite decimals was uncountable. (To include the finite decimals, he converted them to infinite decimals using the fact that a number such as 0.3 can be represented by the infinite decimal 0.29—where the 9s repeat forever.) He used a…

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